Cartridge / Arm Matching....or things you didn't really
want to know about resonant frequency of a system tonearm/cartridge which is
coupled to the record by the compliance of the cantilever suspension. ...

1)

above photo #1: showing the cantilever and location of
suspension. The suspension element may be as simple as a tight fitting elastic
grommet that grips both the shaft and the cartridge body.

2)

above photo #2: Showing stylus mounting in the cantilever. This
is a nude mounting that features a precise 'force fit' between stylus body and
the bore in the cantilever. Additionally, a small bit of adhesive is
applied over the joint to ensure a lasting fit.

side note: Whoever said "diamonds are forever" was
not referring to phono cartridge stylii. They wear out. Yet, like their jewelry
counterparts,
they are expensive nonetheless.

Matching a cartridge to a tonearm (pivoting tone arms)

The focus here is upon elements of the cartridge design, and
elements of the tonearm design which affect how a vinyl addict might go about
choosing a given cartridge to fit a given tonearm. First, some
definitions.

Compliance: Let's take a look at a spring (any spring) designed to carry a given
load. Place too much weight on this spring and it collapses. Place
not enough weight on the spring and the spring won't compress at all and remains
rigid. Think of the cantilever as a spring.

In the case of a phonograph cartridge, the cantilever is a rigid arm connected to a springing medium mounted up within
the body of the cartridge. This springing element may be as simple as a
rubber donut that holds enough tension against the cantilever mounting to
maintain relative position of the cantilever. Most importantly, the
suspension must control the attitude of the stylus fitted at the cantilever's end.

The amount of distance that a cantilever deflects under a given force
load is referred to as
cantilever compliance. Higher compliance cantilevers deflect a greater
distance when a given amount of force is applied. Lower compliance
cantilevers deflect lesser distances when the same force is applied. In
other words; high compliance = softer, low compliance = stiffer.

Vertical Tracking Force (VTF): The amount of vertical force placed upon the
cantilever is controlled by a careful balance between the weights of the arm on
either end of the pivot. Typically, the tonearm will have a long tube
leading to a head shell with a phonograph cartridge and stylus at that
end. The other end of the tonearm will have weights intended to balance
the mass of the long end. By careful manipulation of the counterweight
position, relative to the distance from the pivot bearing, a precisely
measured amount of vertical force may be applied at the stylus end.

Effective Mass: The amount of force felt at the
stylus under dynamic conditions in any (xyz) arcing vector about the pivot. This
differs from VTF which is set static and remains constant only under 'peaceful'
conditions while the record is in play. Effective mass is influenced by
the weight of the various appendages of the tonearm assembly in ratio to the
distance from the pivot. Weight that is further from the pivot center will
account for higher effective mass than the same weight if positioned closer to
the pivot. Said slightly differently, the heavy bits on the tonearm
need to be closer to the tonearm pivot or excessive effective mass will be the
result.

Phonograph cartridges have different weights from one product to the next,
therefore tonearm makers rate their arms in terms of effective mass before a
cartridge is mounted. In the case of the SME 3009 Improved with fixed head
shell, the arm is rated to have an effective mass of 6.5 grams. Compare
this to the Thorens TP16 Mk 1 tonearm which has a rated
effective mass of 16.5grams.

Resonant
frequency of the cantilever is used as a guide to match suspension stiffness
(compliance) of the cartridge to the tone arm's (effective) mass.

Resonant Frequency (of the cantilever) The acoustic frequency at which the cantilever
will become excited and vibrate out of control. :))) This frequency is
measured in cycles per second. Also referred to as 'hz'. Resonant
frequency of a cantilever is regarded as inescapable and the effect is
controlled by manipulating this frequency to exist in a range below human hearing
but not so low that
it will become excited by external vibrations such as foot fall disturbance or
that of a warped record. This ideal frequency
range is 8 to 12 hz. The lowest of low organ notes rarely go
below 20 hz. Footfall and record warps happen below 6 hz.

The effective mass of a tonearm in combination with the compliance of the
cartridge cantilever serves to determine where the resonant frequency of a given
tonearm/cartridge match up will be.In general terms, arms with high
effective mass fitted with cartridges of high compliance result in resonant
frequencies that fall below the ideal range. At the opposite end, arms
with low effective mass mated to cartridges of low compliance result in resonant
frequencies above the desired range. Both extremes are to be
avoided.

So....wouldn't it be useful if you could calculate a
resonant frequency between a given arm and cart...? You
can, just plug in some vital statistics into the short formula below.
Keep in mind the formula is intended as a rule of thumb. The test record
will be the final say on what a given arm cart combo can give in terms of
cantilever resonant frequency. Think of the formula as a 'on paper test'
and the test record as a 'real world actual test'.

Example No. 1: Now let us think about matching a
Shure V15VxMR to the above pictured TP16 Mk 1 tonearm.

effective mass rating: 16.5 grams

cartridge weight: 6.6 grams

fastener weight: .5 grams

compliance: 25

(16.5 + 6.6 + .5) * 25 = 590

sqrt 590 = 24.2899

159 / 24.2899 = 6.5459 hz calculated

The above figure was approximately verified with the HFNRR test record
getting a test result value of 6 hz

I'll admit that I have lived with the above combination for a time. I found
the setup susceptible to footfall but it seemed to track record warps just
fine. A marginal matching. It is the wrong side of marginal, too.
Another visual note about this match-up is that the high compliance of the Shure
cantilever was quite obvious when dropping the stylus into the lead in
groove. Considerable sideways deflection was evident. This is not
the same case with the above match up between the Shure and the much lighter SME
tonearm. For more notes about The Shure cartridge and it's dynamic
stabilizer when fitted to the TP16
mk 1 see the ** below.

Example No. 2: with the TP16 Mk 1 tonearm.
This time with the above pictured Blue Point Special cartridge...? Let's
crunch the numbers and see.

Notes about the discrepancy between the test record figure and the
calculation. The test record value is considered to be the valid
reference. One of the variables in the calculation must be
incorrect. Perhaps effective mass rating of the tonearm may actually be
lighter in the real world situation. The finger lift was not used and so
does not contribute to the effective mass. I have no accurate means to
measure the weight of the finger lift. I used the shortest possible screws
in the cartridge mounting. Said screws and nuts are aluminum.

How to calculate Compliance based on a Test
record result for resonant frequency:

*Shure does not publish compliance figures for this cartridge in the owners
manual that comes with it. A search of the knowledge base at the Shure
website turned up the following information on the V15VxMR:

** : The Shure V15VxMR phono cartridge uses a device that Shure
calls a "dynamic stabilizer". They (Shure) describe this device
as a shock absorber for their cantilever. If this device is put to use, it
is claimed that a much wider range of effective mass tone arms may be used with
this cartridge. I have used this 'dynamic stabilizer' when the cartridge
was fitted to the heavy TP16 mk1 and found that the cartridge tracked all
records without any apparent fault. When the device was parked in it's
'up' position (taken out of use) a much greater amount of cantilever deflection
could be witnessed and the tonearm was more susceptible to external
disturbance. The 'device' was not put to use with the SME tonearm as the
resonant frequency of the cantilever falls within the optimal range when fitted
to this arm.

*** : All math/physics formulas and explanations offered on this page were either collected
from or donated by persons more capable and knowledgeable than myself. I
just operate them. If you see an error or have another formula useful to
this topic please contact me.